There is proposed a cryptographic system that realizes sophisticated cryptography or a signature by using a pairing operation on a composite order.                [Non-patent Literature 1] D. Boneh, A. Sahai, B. Waters, “Fully Collusion Resistant Traitor Tracing With Short Ciphertexts and Private Keys”, Eurocrypt 2006, 2006        [Non-patent Literature 2] D. Boneh, B. Waters, “A Fully Resistant Broadcast, Trace, and Revoke System”, ACM CCS 2006, 2006        [Non-patent Literature 3] D. Boneh, E. Goh, K. Nissim, “Evaluating 2-DNF Formulas on Ciphertexts”, TCC 2005, 2005        [Non-patent Literature 4] X. Boyen, B. Waters, “Full-Domain Subgroup Hiding and Constant-Size Group Signatures”, PKC 2007, 2007        [Non-patent Literature 5] D. Boneh, B. Waters, “Conjunctive, Subset, and Range Queries on Encrypted Data”, TCC 2007, 2007        [Non-patent Literature 6] S. D. Galbraith, J. Pujolas, C. Ritzenthaler, B. Smith, “Distortion Maps For Genus Two Curves”, arXiv: math.NT/0611471, 2006        [Non-patent Literature 7] S. D. Galbraith, F. Hess, F. Vercauteren, “Hyperelliptic pairings”, Pairing 2007, LNCS 4575, pp. 108-131, 2007        [Non-patent Literature 8] D. Freeman, “Constructing Pairing-Friendly Genus 2 Curves with Ordinary Jacobians”, Pairing 2007, LNCS 4575, pp. 152-176, 2007        